If there was no λ and μ, I think I'd be able to reasonably solve this. How do I show that these real numbers don't matter in this exchange?

There is really no difference between showing it with a scalar multiple and without the scalar multiple, since the scalars distribute to every term in the matrix. You could just as easily let ##A_\lambda = \lambda A## and ##B_\mu = \mu B## and show what you say you could show for the new matrices. Then the last step is to show that you can pull the scalars back out...since they are distributed to each term, they will not affect the solution.

Lc and Ruber, thanks for the words of advice. What you stated resonates well with me. I will be working on this problem in the next 24 hours and I feel confident about it. I will post back with my update. :)